## Vehicle Routing Problem for Multiple Product Types, Compartments, and Trips with Soft Time Windows

Read this article. The paper presents a mathematical model to solve vehicle routing challenges. In your own words, explain what vehicle routing is and why it's important to a supply chain.

### Problem Formulation

In this section, we present a mathematical
model for the vehicle routing problem for multiple product types,
compartments, and trips with soft time windows. The following notations
are introduced to formulate the mathematical model.

Sets (Indexes). Let be
a directed graph where is a set of vertices and is a set of arcs
representing the connections between the depot and customers and among
the customers, where . Indeed, let

be a node set (o is origin and d is destination),

be a set of compartments,

be a set of trips.

*Parameters. Let *

be
a distance between customers and ,

be a time (hour) from customers to
(e.g., when means traveling speed of each vehicle),

be the demand of
the customer in product ,

be the capacity of a compartment in the
vehicle ,

be an arbitrary big number (e.g., ),

be the fixed cost per
kilometer of running of each vehicle,

be the cost per unit travel per
unit of load of product type by the vehicle,

be a service time at the
customer ,

be the lower bound of soft time window for the node ,

be
the upper bound of soft time window for the node ,

be the lower bound
of hard time window for the node ,

be the upper bound of hard time
window for the node ,

be penalty cost per hour for the earliness,

be penalty cost per hour for the lateness.

##### Decision Variables. Consider

is the quantity of product type to be transported from node to node by
vehicle in trip ,

is the quantity of product type to be disposed at
node by vehicle in trip ,

is the starting time of service in node by the vehicle in trip ,

is the violation degree of a lower bound of soft time window,

is the violation degree of an upper bound of soft time window, and

The problem is solved under the following assumptions:

(1) Every customer can be served by one and only one vehicle.(2) Every vehicle starts and ends only at the central depot.

(3) The total demand on one particular route must not exceed the capacity of the vehicle.

(4) Each customer must be served in the related hard time window.

(5) A service (wait) time in each customer is allowed.

(6) The hard time windows must not be violated.

(7) The soft time windows can be violated at a fixed cost.

*Mathematical Model. *The
vehicle routing problem for multiple product types, compartments, and
trips (model 1) can then be stated mathematically as follows:

We
minimize objective function (3) which consists of two components. The
first component represents traveling cost. It depends on distances
covered by vehicles and the fixed costs per unit distance. The second
component is defined as the fleet cost which is equal to the sum of the
costs related to the loading and distance. Constraints (4) are the flow
conservation of a vehicle. That is, for , every vehicle starts only once
at the depot and no arc terminates at node , and also, for , every
vehicle ends only once at the depot and no arc originates from node ,
and indicate that exactly one vehicle enters and leaves each customer
node. Constraints (5) require the fact that each customer is serviced
exactly once. Constraints (6) state that all customers must be assigned
to at most one vehicle. Constraints (7) ensure that the total number of
quantity transported does not exceed the vehicle's capacity. Constraints
(8) state that the sum of the quantities of product type disposed at a
customer by all the vehicles must be equal to the demand for product .
Constraints (9) represent the sum of quantities of the product type to
be transported by the vehicle to the customer from the depot node (o). It
should in total be greater than the total sum of the demands at all
customers. Constraints (10) state that the quantity of product
transported to a customer minus the quantity disposed at that customer
must be equal to the quantity transported to the next customer by the
vehicle. Constraints (11) represent the transported quantity. That is,
variable if route segment is not used by vehicle ; otherwise it should
not exceed the total tonnage capacity of vehicle . Constraints (12)
state that each vehicle must not load at the depot. Constraints (13) and
(14) are the flow conservation constraints which describe the vehicle
path. Constraints (15) define a limit for each compartment to be
dedicated to a single type of product.

This model can be extended
to the model called vehicle routing problem for multiple product types,
compartments, and trips with time windows (model 2). In the next model,
we include the time windows for each node. Consider

(16)

Constraints (16) ensure feasibility of the time schedule; that is,

must be greater than or equal to \(\) whenever the vehicle travels from to . By rewriting constraints (16), they can be replaced with the following constraints:

Constraints (18) ensure that the starting time of a service does not violate the given hard time window.

Proposition 1. Constraints

are valid for model 2. Moreover, any that satisfies constraints (19) satisfies constraints (17).

Proof of Proposition 1 is straightforward and, thus, we omit it.

The vehicle routing problem for multiple product types, compartments, and trips with time windows can be stated as follows:

subject to: (4) - (15), (18), (19)

Constraints
(18) assure that the starting time of services does not violate the
given hard time windows. Constraints (21) assure that the starting time
in next trip of each vehicle is satisfied.

Finally we consider
the complete the generalization of all models mentioned in this paper.
We include soft time windows for each time window allowing the vehicle
to enter the depot earlier or later with some penalties. The vehicle
routing problem for multiple product types, compartments, and trips with
soft time windows (model 3) can then be stated mathematically as

subject to: (4) - (15) , (22),

The
objective function differs from models 1 and 2 by adding the third
component. The objective function is to minimize the sum of the cost of
traveled distances, total of costs related to loading products, and the
total penalties of the outrage from soft time windows. We then replaced
constraints (19) and (18) by constraints (23). Constraints (24)
determine the violation degree of the lower and upper bound of soft time
windows for each node.